SYMMETRY Finite geometric objects (“molecules) are symmetric with respect to a) rotations b) reflections (symmetry planes, symmetry points) c) combinatons of translations and rotations (screw axes) 2π If an object is symmetric with respect to rotation by angle φ = , it is said to have has an “n-fold rotational axis” n n=1 is a trivial case: every object is symmetric about a full revolution. Non-trivial axes have n>1. 1 Symmetries of an equilateral triangle 2 2 2 180-rotations (flips) 3’ 1’ about 22’ 3 1 3 1 2’ about 33’ about 11’ 1 3 1 2 2 3 3 2 2π 1 120 degree= rotations 3 3 2 1 3 3 3 4 2 2 1 1 An equilateral triangle is not symmetric with respect to inversion about its center. But a square is. 5 Gliding (screw) axis Rotate by 180 degrees about the axis Slide by a half-period 6 2-fold rotations can be viewed as reflections in a plane which contains a 2-fold rotation axis 7 Crystal Structure Crystal structure can be obtained by attaching atoms or groups of atoms --basis-- to lattice sites. Crystal Structure = Crystal Lattice + Basis Crystal Structure 88 Partially from Prof. C. W. Myles (Texas Tech) course presentation Building blocks of lattices are geometric figures with certain symmetries: rotational, reflectional, etc. Lattice must obey all these symmetries but, in addition, it also must obey TRANSLATIONAL symmetry Which geometric figures can be used to built a lattice? 9 10 Crystallographic Restriction Theorem: Crystal lattices have only 2-,3, 4-, and 6-fold rotational symmetries 11 B’ ma A’ n-fold axis φ-π/2 φ=2π/n a A B 1. Suppose that an n-fold rotational axis is going through A. another n-fold axis is going through B. 2. Rotate the lattice about A by φ=2π/n: BB’. 3. Rotate the lattice about B by φ=2π/n: AA’. 4. Distance A’B’ must be an integer multiple of a. 12 B’ ma A’ n-fold axis φ-π/2 φ=2π/n a A B ma = a + 2asin(ϕ − π / 2) = a − 2acosϕ 1− m −1 ≤ cosϕ = ≤ 1; m ≥ 0 ⇒ 0 ≤ m ≤ 3 2 m = 0 ⇒ cosϕ=1/2 ⇒ϕ = π / 3 ⇒ n = 6 m = 1⇒ cosϕ=0 ⇒ϕ = π / 2 ⇒ n = 4 m = 2 ⇒ cosϕ=-1/2 ⇒ϕ = 2π / 3 ⇒ n = 3 m = 3 ⇒ cosϕ=-1⇒ϕ = π ⇒ n = 2 13 Icosahedron Penrose tiling 14 Quasicrystals • Discovered in 1984 in Al-Mn submicron granules • Have long-range five-fold orientational but no translational long-range order • Exhibit sharp diffraction patterns • Hard metals with low heat conductivity • A commercially produced Al–Cu–Fe–Cr thermal spray cookware coating • Hydrogen storage? 15 Dan Shechtman Nobel Prize in Chemistry 2011 2 mkm grains Al-Mn, Fe, Cr (10-14%) 16 Fig. 4 The fivefold (A), threefold (B), and twofold (C) diffraction patterns obtained from a region (red dashed circle) of the granule in Fig. CuAl, CuAl2 CuAl_2,CuAl L Bindi et al. Science 2009;324:1306-1309 17 Published by AAAS Simplest type of lattices: Bravais lattices Two equivalent definitions: A. An infinite array of discrete points with an arrangement and orientation that appear exactly the same, from whichever of the points the array is viewed. B. A lattice consisting of all points with positions vectors R of the form R = n1a1 + n2a2 + n3a3 Collorary to B: every point of a Bravais lattice can be reached from any other point by a finite number of translations. Non-Bravais lattice contains points which cannot be reached by translations only. Rotations and reflections must be used in addition to translation. Bravais lattices: basis consists of one element (atom) Non-Bravais lattices: can be represented as Bravais lattices with a basis consisting of more than one element. 18 Bravais lattices in 2D (all sites are equivalent) Oblique Rhombohedral Orthorhombic (Rectangular) 2π 2π 180 = ⇒ 2 − fold axis 180 2π = 180 = 2 2 2 2π 60 = ⇒ 6-fold axis 6 2π 120 = ⇒ 3-fold axis 3 2π 90 = ⇒ 4-fold axis Tetragonal (Square) Hexagonal (Triangular) 19 4 Is this a Bravais lattice? Yes, it is orthorhombic (rectangular) Bravais lattice 20 and the answer is …. NO! 21 “body-centered tetragonal”= tetragonal 22 Bravais lattices in 3D: 14 types, 7 classes Classes: Ag,Au,Al,Cu,Fe,Cr,Ni,Mb… 1. Cubic ✖3 2. Tetragonal✖2 3. Hexagonal✖1 Ba,Cs,Fe,Cr,Li,Na,K,U,V… α − Po 4. Orthorhombic✖4 5. Rhombohedral✖1 6. Monoclinic✖2 7. Triclinic✖1 He,Sc,Zn,Se,Cd… S,Cl,Br F Sb,Bi,Hg 23 Is there a two-dimensional body-centered tetragonal lattice? simple tetragonal body-centered tetragonal? A. Yes B. No 24 An example of a non-Bravais lattice A B Graphene: Honeycomb • Even though graphene is a monoatomic compound, the basis consist of two atoms! • Red(A) and blue (B) sites are not equivalent. 2525 Nobel Prize 2010 Konstantin Novoselov Andre Geim 26 Honeycomb lattice: oblique lattice of diatomic molecules 27 From Ashcroft and Mermin Unit cell: an element of lattice that fills the space under translations. 28 Primitive Cell: The smallest component of the crystal (group of atoms, ions or molecules) that, when stacked together with pure translational repetition, reproduces the whole crystal. S S S S S b S S S S S a S S S S S Primitive unit cell Crystal Structure 29 From Prof. C. W. Myles (Texas Tech) course presentation Blue square=primitive cell (area=1/2) Red parallelogram=unit cell (area=3/4) Green square=unit cell (area=1) Is the blue triangle a unit cell? Crystal Structure 30 From Prof. C. W. Myles (Texas Tech) course presentation NO. Space must be filled by unit cells using only translations. 31 The choice even of a primitive cell is not unique S S’’ b S S’ a Areas: S=S’=S’’ Crystal Structure 32 From Prof. C. W. Myles (Texas Tech) course presentation 2D “NaCl” Crystal Structure 33 From Prof. C. W. Myles (Texas Tech) course presentation Choice of origin is arbitrary - lattice points need not be atoms - but the primitive cell area should always be the same. Crystal Structure 34 From Prof. C. W. Myles (Texas Tech) course presentation - or if you don’t start from an atom Crystal Structure 35 From Prof. C. W. Myles (Texas Tech) course presentation This is NOT a primitive cell - empty space is not allowed! Crystal Structure 36 From Prof. C. W. Myles (Texas Tech) course presentation Graphene lattice as a Rhombohedral Bravais lattice with a 2-atom basis ✖ ✖ ✖ ✖ ✖ ✖ ✖ ✖ ✖ ✖ ✖ ✖ 37 Graphene a2 a1 One possible choice of unit lattice vectors The electronic properties of graphene A. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim Another possible choice of unit lattice vectors Rev. Mod. Phys. 81, 109 (2009) shown along with the unit cell 38 Primitive and conventional cells Primitive cell: • A smallest volume (area) of space that, when translated through all the vectors of a Bravai lattice, fills all space without either overlapping or leaving voids. • Symmetry of the primitive cell does not necessarily coincide with that of a Bravais lattice. • The choice of a primitive cell is not unique • All primitive cells have the same volume (area) Different choices of a primitive cell for an oblique 2D Bravais lattice (from Ashcroft and Mermin) 39 Primitive cells of Bravais lattices contain one atom. If this atom has an odd number of electrons, the element is a metal 40 Basis vectors for a bcc lattice one possible choice of basis a more symmetric choice vectors for a bcc lattice a a a1 = (yˆ + zˆ − xˆ ), a2 = (zˆ + xˆ − yˆ ), a 2 2 a1 = axˆ, a2 = ayˆ, a3 = (xˆ + yˆ + zˆ) a 2 a3 = (xˆ + yˆ − zˆ) 2 41 bcc lattice as a superposition of two simple cubic lattices 42 Basis vectors of an fcc lattice a a a a1 = (yˆ + zˆ), a2 = (zˆ + xˆ ), a3 = (xˆ + yˆ ) 2 2 2 43 Volume of a primitive cell: bcc lattice V = a i a × a 1 ( 2 3 ) a a a a1 = (yˆ + zˆ − xˆ ), a2 = (zˆ + xˆ − yˆ ), a3 = (xˆ + yˆ − zˆ) 2 2 2 2 ⎡ xˆ yˆ zˆ ⎤ 2 ⎛ a⎞ ⎢ ⎥ ⎛ a⎞ V = a1 i(a2 × a3 ) = a1 i 1 −1 1 = a1 i 2(yˆ + zˆ) ⎝⎜ 2⎠⎟ ⎢ ⎥ ⎝⎜ 2⎠⎟ ⎣⎢ 1 1 −1 ⎦⎥ 3 ⎛ a⎞ a3 = 2(yˆ + zˆ − xˆ )i(yˆ + zˆ) = ⎝⎜ 2⎠⎟ 2 44 (conventional) unit cell: • A volume (area) that fills all space, when translated by some lattice vectors. • Larger than a primitive cell • Allows to “see” the lattice better 3D face-centered cubic (fcc) lattice 60 degree rombohedron 45 From Prof. C. W. Myles (Texas Tech) course presentation The most important primitive cell: the Wigner-Seitz cell WS cell about a lattice point: a region of space that is closer to a given lattice point than to any other point. WS cell has the same symmetry as the lattice itself. Simple construction method: connect the lattice points by line, choose the middle points, and draw lines normal to the connecting lines. The enclosed volume (area) is the WS cell. The method of construction is the same as for the Brillouin zone in the wavenumber (reciprocal) space. 46 WS cell for a tetragonal (square) Bravais lattice 47 Construction of a WS cell for a rectangular lattice n.n. n.n.n. n.n.n. n.n. What about next-to-next-to-nearest neighbors? Adding n.n.n.n. does not change the WS cell for a n.n.=nearest neighbor rectangular lattice n.n.n.=next-to-nearest neighbor 48 WS cell for a hexagonal 2D lattice NB: In 2D, WS cells are either hexagons or rectangulars 49 WS cell for an oblique lattice n.n.